Thread Rating:
  • 0 Vote(s) - 0 Average
  • 1
  • 2
  • 3
  • 4
  • 5
[Regular tetration] [Iteration series] norming fixpoint-dependencies
Another picture which shows the wobble of different values for the regular height-function when fp0 or fp1 is used. Example: base b=sqrt(2).

Then the value "normzero" is the map of 1 into the segment seg1 between 2 and 4 using fp0-powerseries and has reference-height 0 in that segment. It is that value of 2.46... = tet0(1,Pi*I/ln(fp0))

To have computations numerically nearer at the fixpointvalue 2, I increase its height (tetrate it using tet0) by 13.5.

Then I generate a set of x-coordinates in small steps in the height-interval hgh0(x)= 13... 17. Now I determine the heights of these x-coordinates using the hgh1()-function which employs the second fixpoint fp1.
Then the height-values using hgh0(x) and hgh1(x) differ periodically by small differences of about 1e-25.

This is the basic idea of the curves in the plot.

But we find, that the norming process has more implications. If we connect the tet0 and tet1-functions using a common x at a fractional iterate from "normzero", then the difference-curve becomes asymmetric.
Examples: if we use the connection-value at tet0(normzero,+0.25) all differences are positive, if at tet0(normzero,+0.5) we have nearly the same curve as with tet0(normzero,0) itself, and if we connect tet0 and tet1 at tet0(normzero,+0.75) all differences become negative.

So the selection of the connection-point for the norming is an important aspect. However, the matter is not yet satisfactorily solved: still we have a small (but seemingly constant ~ 2e-26) difference of the curves for connection-point tet0(normzero,+0) and tet0(normzero,+0.5). So the wobbling is not exact the same even at half-integer steps of the connection-point.


Gottfried Helms, Kassel

Messages In This Thread
RE: [Regular tetration] norming fixpoint-dependencies - by Gottfried - 07/29/2010, 11:36 AM

Possibly Related Threads…
Thread Author Replies Views Last Post
  What are the types of complex iteration and tetration? Daniel 5 321 08/17/2022, 02:40 AM
Last Post: JmsNxn
Question Tetration Asymptotic Series Catullus 18 1,223 07/05/2022, 01:29 AM
Last Post: JmsNxn
Question Formula for the Taylor Series for Tetration Catullus 8 1,582 06/12/2022, 07:32 AM
Last Post: JmsNxn
  Fractional iteration of x^2+1 at infinity and fractional iteration of exp bo198214 17 29,764 06/11/2022, 12:24 PM
Last Post: tommy1729
  Categories of Tetration and Iteration andydude 13 30,188 04/28/2022, 09:14 AM
Last Post: MphLee
  Calculating the residues of \(\beta\); Laurent series; and Mittag-Leffler JmsNxn 0 714 10/29/2021, 11:44 PM
Last Post: JmsNxn
  Trying to find a fast converging series of normalization constants; plus a recap JmsNxn 0 652 10/26/2021, 02:12 AM
Last Post: JmsNxn
  Reducing beta tetration to an asymptotic series, and a pull back JmsNxn 2 1,537 07/22/2021, 03:37 AM
Last Post: JmsNxn
  Perhaps a new series for log^0.5(x) Gottfried 3 5,523 03/21/2020, 08:28 AM
Last Post: Daniel
Question Taylor series of i[x] Xorter 12 25,802 02/20/2018, 09:55 PM
Last Post: Xorter

Users browsing this thread: 1 Guest(s)